Sentences (1) and (3):
Mis amigas quieren que nos reunamos.
Mis amigas quieren que nos encontremos.
Mis amigas quieren que nos reunamos.
- Mis amigas quieren que nos encontremos.
are correct and similar in meaning. They are also equally ambiguous, both in English and in Spanish -- does "us" (or "nos") include "my friends" (or "mis amigas") or not?
(2) is also correct but different from (1) and (3) in that it is not ambiguous:
- Mis amigas quieren reunirnos.
as in this case "nos" does not include my friends. A possible context would be that the girls want some other people (including the speaker) to reunite.
Sentence (4):
- Mis amigas quieren que quedemos.
sounds incomplete to me, except in some very specific context, like "quedar en un concurso" (remain in a competition).
If the intention is to say My friends want us to stay we should remember that this use of the verb is pronominal in Spanish:
- Mis amigas quieren que nos quedemos.
Again, it is unclear whether the first person plural includes "mis amigas" or not.